Power Rule
Derivative of a power function.
Formula
Variables
Example
d/dx x³ = 3x²
How it works
The power rule tells you how the slope of a simple algebraic term x^n changes: you bring the exponent down in front and reduce the exponent by 1. It works because differentiation measures the rate of change, and powers change in a predictable pattern as x changes. This rule is one of the main shortcuts in calculus because it lets you differentiate polynomial terms quickly without going back to the full limit definition each time.
When to use it
Use this whenever you need the derivative of a variable raised to a constant power, such as x^5, x^(1/2), or x^(-3). It is especially useful in algebra, physics, economics, and any problem involving polynomial or power-based models.
Worked example, step by step
Find the derivative of f(x) = x^4 Use the power rule: d/dx(x^n) = nx^(n-1) Here, n = 4 So f'(x) = 4x^(4-1) Simplify the exponent: f'(x) = 4x^3
Common mistakes to avoid
- Forgetting to subtract 1 from the exponent after bringing it down
- Using the power rule on terms that are not a single power of x, such as x^n + 5, without differentiating each term separately
- Treating the exponent as if it were the variable and the base as if it were constant
Did You Know?
The power rule is usually the very first derivative students learn, and it works for any real exponent.
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More in Calculus
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IntermediateDerivative of a power function.
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IntermediateDerivative of a composite function.
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IntermediateDerivative of a product of two functions.
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IntermediateDerivative of a quotient of two functions.
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IntermediateDerivative of the sine function.
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IntermediateDerivative of the cosine function.